1. Introduction
Spiral bevel gears (Gleason type) are crucial components in various mechanical systems, especially in high – speed and heavy – load applications such as automotive, aerospace, and marine industries. Their unique characteristics, including high – transmission efficiency, compact structure, and stable operation, make them irreplaceable in many scenarios. However, due to the complex tooth surface geometry of spiral bevel gears, accurately understanding their behavior during meshing, especially the dynamic changes of tooth – surface contact, is a challenging task. This article aims to comprehensively introduce the parameter modeling and finite – element contact analysis of spiral bevel gears, providing a detailed guide for gear design, manufacturing, and optimization.
1.1 Significance of Spiral Bevel Gears
Spiral bevel gears play a vital role in power transmission systems. In automotive differentials, for example, they enable smooth torque distribution between the left and right wheels while allowing for speed differences during cornering. In aerospace applications, such as aircraft engines and landing gear systems, spiral bevel gears must withstand high loads and operate reliably under extreme conditions. Their performance directly affects the overall efficiency, durability, and safety of the mechanical equipment.
1.2 Challenges in Studying Spiral Bevel Gears
The complex tooth – surface geometry of spiral bevel gears is a major obstacle to in – depth research. Unlike spur gears with relatively simple tooth profiles, the tooth surface of spiral bevel gears is formed through a complex machining process. This complexity makes it difficult to establish accurate mathematical models and analyze the contact behavior between teeth. Additionally, accurately describing the dynamic changes of tooth – surface contact during meshing requires advanced computational methods and experimental techniques.
2. Tooth – Surface Modeling of Spiral Bevel Gears
2.1 Tooth – Surface Equation of the Large Gear
The large gear of spiral bevel gears (right – hand rotation in this case) is usually processed by the generating method. The tooth surface is part of the envelope surface of the cutting tool’s trajectory. During the machining process, the cutter head rotates around its axis to form a cutting cone surface, and at the same time, the cradle and the large – gear blank also rotate around their axes, thus shaping the tooth surface of the large gear.
To establish the tooth – surface equation, we first need to set up coordinate systems. As shown in Table 1, we establish a coordinate system for the cutter head position and another for the gear – blank position.
| Coordinate System | Related Points | Key Parameters |
|---|---|---|
| Cutter – head position coordinate system | \(O_{ml}\) (machine center), \(O_{g1}\) (cutter – head center) | \(S_{R}\) (axial cutter position), \(Q_{ra}\) (angular cutter position) |
| Gear – blank position coordinate system | \(O_{m2}\) (design cone vertex of the gear – blank) | \(X_{B}\) (bed position), \(X_{1}\) (axial gear position), \(\delta_{1}\) (gear – blank installation angle) |
2.3 Tooth – Surface Equation of the Small Gear
The small gear (left – hand rotation) of spiral bevel gears is processed by the tool – tilting method. The processing principle is basically similar to that of the large gear. The cutter – head position and gear – blank position are shown in Figure 2. [Insert Figure 2: Cutter – head and gear – blank positions for the small gear]
The tooth surface of the small gear processed by the tool – tilting method is enveloped by the cutting cone surface of the cutter head. During the meshing process, according to the meshing equation at the contact point, the relative velocity \(v_{12}\) between the cutter head and the gear – blank is perpendicular to the normal vector \(n_{2}\), that is \(v_{12}\cdot n_{2} = 0\). Finally, we can obtain the tooth – surface equation \(r_{3}(\theta_{g},U_{g})\) of the small gear. The boundary conditions and solution methods are similar to those of the large – gear tooth surface.
3. Modeling of Spiral Bevel Gears
3.1 Parameter Input
To build a three – dimensional model of spiral bevel gears, we need to input the basic parameters of the gear pair and the processing parameters. The basic parameters of the gear pair are shown in Table 2.
| Parameter | Large Gear | Small Gear |
|---|---|---|
| Number of Teeth | 33 | 28 |
| Face Module (mm) | 4.8 | 4.8 |
| Outer Cone Distance (mm) | 123.3 | 123.3 |
| Tooth Width (mm) | 37 | 37 |
| Addendum Height (mm) | 2.2 | 2.3 |
| Dedendum Height (mm) | 3.3 | 2.9 |
| Pitch – Cone Angle (°) | 49.4 | 40.2 |
| Root – Cone Angle (°) | 47.5 | 39.3 |
| Face – Cone Angle (°) | 52.5 | 43.1 |
The processing parameters are shown in Table 3.
| Parameter | Small Gear (Concave) | Small Gear (Convex) | Large Gear (Concave) | Large Gear (Convex) |
|---|---|---|---|---|
| Cutter – head Pressure Angle (°) | 20 | 20 | 20 | 20 |
| Cutter – tip Circle Radius (mm) | 1.8 | 2.2 | ||
| Cutter – top Distance (mm) | 2.4 | |||
| Radial Cutter Position (mm) | 93.9 | 93.8 | 92.9 | |
| Angular Cutter Position (°) | 48.4 | 45.5 | 49.3 | |
| Machine – tool Roll Ratio | 2.2 | 2.3 | 1.3 | |
| Bed Position (mm) | 0.55 | – 0.65 | 0 |
3.2 Generation of Discrete Points
We write the tooth – surface equations of the large and small gears into corresponding programs and substitute the above – mentioned parameters into the programs. According to the formula for the large – gear tooth – surface equation, we can calculate the discrete points on the convex and concave surfaces of the large gear, as shown in Figure 3. [Insert Figure 3: Discrete points on the large – gear tooth surface (a) Convex points; (b) Concave points]
3.3 Three – Dimensional Modeling
The obtained discrete – point data is imported into the three – dimensional modeling software SolidWorks in text format. We use the point cloud to perform three – dimensional modeling. By using commands such as curves and surfaces, we can complete the modeling process. For the large gear, we generate the concave and convex surfaces of the large gear by placing the discrete points in the same coordinate system. Then, through surface trimming and intersecting – surface cutting, we can generate the tooth profile of the large gear. Finally, by arraying the tooth profile, we can establish the three – dimensional model of the large spiral bevel gear, as shown in Figure 4. [Insert Figure 4: Process of establishing the three – dimensional model of the large gear (a) Concave and convex surfaces; (b) Surface trimming; (c) Tooth profile; (d) Array formation]
Similarly, we can establish the three – dimensional model of the small gear. After that, we assemble the gear pair according to the specified position using the gear – mating command in the mechanical – mating function of SolidWorks. We can also use the interference – checking function in SolidWorks to check whether the assembly of the spiral bevel gears is reasonable, avoiding interference and providing good mating conditions for the subsequent finite – element dynamic – contact analysis.
4. Finite – Element Contact Analysis of Spiral Bevel Gears
4.1 Establishing the Finite – Element Analysis Model
We use the finite – element software ANSYS Workbench platform and its transient – dynamics module to conduct dynamic – contact analysis of spiral bevel gears. The main steps include setting the model material parameters, meshing, setting the boundary conditions and contact relationships, and setting the solution parameters.
4.1.1 Setting Material Parameters
We import the assembled model into the finite – element software Workbench. According to the actual material of spiral bevel gears (18Cr2Ni4WA), we set the material parameters. The elastic modulus \(E = 202\) GPa, Poisson’s ratio \(\mu=0.273\), and density \(\rho = 7.91\) g/cm³.
4.1.2 Meshing
The accuracy and type of meshing directly affect the accuracy of the three – dimensional model analysis results. If the mesh is too dense, it will increase the calculation time and occupy a large amount of computer resources. To obtain accurate calculation results and improve the calculation efficiency, we choose tetrahedral meshing for the model. We also locally refine the expected meshing contact surfaces of the gears. The refined part has a mesh size of 1 mm, and the non – refined part has a mesh size of 10 mm. The finite – element mesh model is shown in Figure 5, with a total of 619,607 nodes and 418,275 elements. [Insert Figure 5: Finite – element mesh model of spiral bevel gears]
4.1.3 Setting Boundary Conditions and Contact Relationships
During the operation of the spiral – bevel – gear pair, the motor applies a torque that rotates around the central axis to the driving gear through the shaft. The driving gear meshes with the driven gear, driving the driven gear to rotate around its central axis. The driven gear not only has an input torque but also a load. Therefore, the applied load is that the driving gear rotates around the central axis at a speed of 600 r/min. To compare the impact of the load on the contact area, the resistance torques of the driven gear are set to 30 N·m and 500 N·m respectively.
We select the locally refined tooth surface as the contact area, set the large gear as the Contact and the small gear as the Target, and choose the augmented Lagrangian method as the contact algorithm. Considering that the contact area has a frictional force, the friction coefficient is selected as 0.2 under normal working conditions.
4.1.4 Solution Setting
The parameter setting in the transient – dynamics contact analysis is very important. The parameter setting of the load step determines whether the non – linear solution can proceed smoothly. After analysis and testing, the total time of the load step is set to 0.0125 s, the initial load sub – step is 50, the minimum load sub – step is 20, the maximum load sub – step is 3000, the large – deformation option is turned on, and the iterative algorithm is used to perform the transient solution for the non – linear contact.
4.2 Simulation Results Analysis and Experimental Comparison
4.2.1 Simulation Results Analysis
From the simulation results, we can obtain the stress distribution values on the tooth surface during the meshing process and the contact – impression area. According to the actual working conditions, during the operation of spiral bevel gears, generally two or more pairs of teeth are meshing simultaneously. For the convenience of observing the change of the stress – area, we select the simulation results of one pair of tooth surfaces under two different loads (30 N·m and 500 N·m) for explanation. When the driving gear rotates clockwise, we take the starting, middle, and exiting states, and obtain the stress distribution of the working – concave – surface instantaneous contact area. The tooth – surface stress cloud diagrams are shown in Figure 6. [Insert Figure 6: Tooth – surface stress cloud diagrams (a) 30 N·m load; (b) 500 N·m load]
As can be seen from the stress cloud diagrams, with the rotation of the spiral – bevel – gear pair, the tooth – surface contact area moves from the small end to the large end of the tooth. When the load increases, the contact – area size also increases, but the overall contact trend does not change. The contact area is distributed in the middle – small – end part of the tooth surface and shows a jujube – pit shape, forming a certain angle with the tooth – surface direction. Through unit – mesh calculation, the length of the contact area is about 47% of the tooth length, and the height is about 60% of the total tooth height. The simulation results are basically consistent with the design results, indicating that the established model is accurate and reliable.
4.2.2 Experimental Comparison
To further verify the correctness of the model and simulation results, we conduct an actual mating experiment on the ground spiral bevel gears. We use a 500 – mm universal rolling inspection machine, as shown in Figure 7.
Since the universal rolling inspection machine is used to check the overall contact – impression distribution of the contact area, we can achieve the experimental purpose by applying an appropriate load. Here, the load is set to 30 N·m.
We adjust the installation position of the spiral – bevel – gear pair by coloring the tooth surface to ensure normal mating. Then we start the machine. After the tooth surfaces are normally meshed and run for a certain time, we observe the tooth – surface contact situation. The contact impression of the large – gear concave surface is shown in Figure 8.
By analyzing the contact impressions in Figure 6 and Figure 8, we find that the finite – element simulation results are basically consistent with the experimental results. This verifies the accuracy of the established model and the finite – element contact – analysis results. The contact impressions obtained through simulation and experiment both meet the distribution range of the contact area in the design criteria, indicating that the designed and manufactured spiral bevel gears can meet the actual use requirements.
5. Conclusion
This article comprehensively studies the parameter modeling and finite – element contact analysis of spiral bevel gears. By analyzing the forming principle of spiral bevel gears, we solve the discrete points on the tooth surface and establish a three – dimensional model through the tooth – point cloud, which improves the modeling accuracy.
We use the transient – dynamics module in ANSYS Workbench to establish a finite – element model and extract the contact impressions of the large – gear concave surface during the meshing process of spiral bevel gears. The analysis shows that the simulation results are accurate and reliable.
The contact impressions obtained through simulation and experiment meet the requirements of the contact area in the design criteria, indicating that the designed and manufactured spiral bevel gears can meet the actual use requirements. This research provides a reliable basis for the design, manufacturing, transmission – matching, and optimization analysis of spiral bevel gears. Future research can focus on further improving the accuracy of the model, considering more complex working conditions, and exploring new manufacturing processes to enhance the performance of spiral bevel gears.
